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<div class="title">SorSolvers.py</div>  </div>
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<div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">#!/usr/bin/python</span>
<a name="l00002"></a>00002 <span class="comment"># -*- coding: utf-8 -*-</span>
<a name="l00003"></a>00003 
<a name="l00004"></a>00004 <span class="comment"># Copyright (c) 2011</span>
<a name="l00005"></a>00005 <span class="comment">#</span>
<a name="l00006"></a>00006 <span class="comment"># Permission is hereby granted, free of charge, to any person obtaining a</span>
<a name="l00007"></a>00007 <span class="comment"># copy of this software and associated documentation files (the &quot;Software&quot;),</span>
<a name="l00008"></a>00008 <span class="comment"># to deal in the Software without restriction, including without limitation</span>
<a name="l00009"></a>00009 <span class="comment"># the rights to use, copy, modify, merge, publish, distribute, sub license,</span>
<a name="l00010"></a>00010 <span class="comment"># and/or sell copies of the Software, and to permit persons to whom the</span>
<a name="l00011"></a>00011 <span class="comment"># Software is furnished to do so, subject to the following conditions:</span>
<a name="l00012"></a>00012 <span class="comment">#</span>
<a name="l00013"></a>00013 <span class="comment"># The above copyright notice and this permission notice shall be included in</span>
<a name="l00014"></a>00014 <span class="comment"># all copies or substantial portions of the Software.</span>
<a name="l00015"></a>00015 <span class="comment">#</span>
<a name="l00016"></a>00016 <span class="comment"># THE SOFTWARE IS PROVIDED &quot;AS IS&quot;, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR</span>
<a name="l00017"></a>00017 <span class="comment"># IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,</span>
<a name="l00018"></a>00018 <span class="comment"># FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE</span>
<a name="l00019"></a>00019 <span class="comment"># AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER</span>
<a name="l00020"></a>00020 <span class="comment"># LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,</span>
<a name="l00021"></a>00021 <span class="comment"># OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE</span>
<a name="l00022"></a>00022 <span class="comment"># SOFTWARE.</span>
<a name="l00023"></a>00023 <span class="comment">#</span>
<a name="l00024"></a>00024 <span class="comment"># Author: Jesus Carrero &lt;j.o.carrero@gmail.com&gt;</span>
<a name="l00025"></a>00025 <span class="comment"># Mountain View, CA</span>
<a name="l00026"></a>00026 <span class="comment">#</span>
<a name="l00027"></a>00027 
<a name="l00028"></a>00028 <span class="keyword">import</span> numpy <span class="keyword">as</span> np
<a name="l00029"></a>00029 <span class="keyword">from</span> LCSolverBase <span class="keyword">import</span> LCSolverBase
<a name="l00030"></a>00030 <span class="keyword">from</span> scipy <span class="keyword">import</span> diag
<a name="l00031"></a>00031 
<a name="l00032"></a>00032 
<a name="l00033"></a><a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html">00033</a> <span class="keyword">class </span><a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html">Sor</a>(LCSolverBase):
<a name="l00034"></a>00034 
<a name="l00035"></a>00035     <span class="stringliteral">&quot;&quot;&quot;</span>
<a name="l00036"></a>00036 <span class="stringliteral">    Synopsis:</span>
<a name="l00037"></a>00037 <span class="stringliteral">    ---------</span>
<a name="l00038"></a>00038 <span class="stringliteral">      The method is guaranty to converge for symmetric matrices and diagonal dominant matrices</span>
<a name="l00039"></a>00039 <span class="stringliteral"></span>
<a name="l00040"></a>00040 <span class="stringliteral">      Solves   z^T(Az+b) = 0</span>
<a name="l00041"></a>00041 <span class="stringliteral">                   Az+b &gt;= 0</span>
<a name="l00042"></a>00042 <span class="stringliteral">                   z    &gt;= 0</span>
<a name="l00043"></a>00043 <span class="stringliteral"></span>
<a name="l00044"></a>00044 <span class="stringliteral">      SOR          approximate the solution of the linear system Ax = b by applying</span>
<a name="l00045"></a>00045 <span class="stringliteral">                   the SOR method (successive over-relaxation)</span>
<a name="l00046"></a>00046 <span class="stringliteral"></span>
<a name="l00047"></a>00047 <span class="stringliteral">           calling sequences:</span>
<a name="l00048"></a>00048 <span class="stringliteral">                   x = sor ( M, b, xold, omega, TOL, Nmax )</span>
<a name="l00049"></a>00049 <span class="stringliteral">                   sor ( M, b, xold, omega, TOL, Nmax )</span>
<a name="l00050"></a>00050 <span class="stringliteral"></span>
<a name="l00051"></a>00051 <span class="stringliteral">           inputs:</span>
<a name="l00052"></a>00052 <span class="stringliteral">                   M       coefficient matrix for linear system - must be a</span>
<a name="l00053"></a>00053 <span class="stringliteral">                           square matrix, sparse or dense</span>
<a name="l00054"></a>00054 <span class="stringliteral">                   b       right-hand side vector for linear system</span>
<a name="l00055"></a>00055 <span class="stringliteral">                   xold    vector containing initial guess for solution of</span>
<a name="l00056"></a>00056 <span class="stringliteral">                           linear system</span>
<a name="l00057"></a>00057 <span class="stringliteral">                   omega   relaxation parameter</span>
<a name="l00058"></a>00058 <span class="stringliteral">                   TOL     convergence tolerance - applied to maximum norm of</span>
<a name="l00059"></a>00059 <span class="stringliteral">                           difference between successive approximations</span>
<a name="l00060"></a>00060 <span class="stringliteral">                   NMax    maximum number of iterations to be performed</span>
<a name="l00061"></a>00061 <span class="stringliteral"></span>
<a name="l00062"></a>00062 <span class="stringliteral">           output:</span>
<a name="l00063"></a>00063 <span class="stringliteral">                   x       approximate solution of linear system</span>
<a name="l00064"></a>00064 <span class="stringliteral"></span>
<a name="l00065"></a>00065 <span class="stringliteral">        Reference:</span>
<a name="l00066"></a>00066 <span class="stringliteral">            [0] S. Ikonen and J. Toivanen. Efficient numerical methods for pricing American op-</span>
<a name="l00067"></a>00067 <span class="stringliteral">            tions under stochastic volatility.</span>
<a name="l00068"></a>00068 <span class="stringliteral">    &quot;&quot;&quot;</span>
<a name="l00069"></a>00069 
<a name="l00070"></a>00070     __init__ = [<span class="stringliteral">&#39;m_omega&#39;</span>]
<a name="l00071"></a>00071 
<a name="l00072"></a>00072     <span class="keyword">def </span>__init__(self, M, b, method=None, omega=None, constraint=None):
<a name="l00073"></a>00073         LCSolverBase.__init__(self, M, b, method, constraint)
<a name="l00074"></a>00074         <span class="keyword">assert</span> omega == <span class="keywordtype">None</span> <span class="keywordflow">or</span> (0 &lt; omega <span class="keywordflow">and</span> omega &lt;= 2)
<a name="l00075"></a>00075         self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a> = omega
<a name="l00076"></a>00076 
<a name="l00077"></a>00077     <span class="keyword">def </span>_relaxationParameter(self):
<a name="l00078"></a>00078         M = self.get_matrix()
<a name="l00079"></a>00079         diags = diag(M)
<a name="l00080"></a>00080         rho = 0.
<a name="l00081"></a>00081         (nRows, cols) = M.shape
<a name="l00082"></a>00082         <span class="keywordflow">for</span> i <span class="keywordflow">in</span> range(nRows):
<a name="l00083"></a>00083             row = abs(M[i, :])
<a name="l00084"></a>00084             row[i] = 0.
<a name="l00085"></a>00085             <span class="keyword">assert</span> 0 != diags[i]
<a name="l00086"></a>00086             rho = max(rho, sum(row) / diags[i])
<a name="l00087"></a>00087         <span class="keywordflow">return</span> 2. / (1. + np.sqrt(1. - rho ** 2))  <span class="comment"># Ref [0]</span>
<a name="l00088"></a>00088 
<a name="l00089"></a>00089     <span class="keyword">def </span>solve_lc_tridia(self):
<a name="l00090"></a>00090         <span class="keywordflow">if</span> <span class="keywordtype">None</span> == self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a>:
<a name="l00091"></a>00091             self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a0bd7ed48c6a7a39e945be4009251b731">omega</a> = self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a10bb69b8ef69ccea498ffbc372f95df6">_relaxationParameter</a>()
<a name="l00092"></a>00092 
<a name="l00093"></a>00093         M = self.get_matrix()
<a name="l00094"></a>00094         diag0 = diag(M)
<a name="l00095"></a>00095         (diag1, diag_1) = (diag(M, 1), diag(M, -1))
<a name="l00096"></a>00096 
<a name="l00097"></a>00097         b = self.get_rhs()
<a name="l00098"></a>00098         xn = self.solution()
<a name="l00099"></a>00099 
<a name="l00100"></a>00100         const = self.get_constraint()
<a name="l00101"></a>00101         <span class="keywordflow">if</span> <span class="stringliteral">&#39;pSor&#39;</span> == self.get_lcp_solver_type():
<a name="l00102"></a>00102             <span class="keyword">assert</span> const != <span class="keywordtype">None</span>
<a name="l00103"></a>00103         <span class="keywordflow">else</span>:
<a name="l00104"></a>00104             const = xn
<a name="l00105"></a>00105 
<a name="l00106"></a>00106         omega = self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a>
<a name="l00107"></a>00107         xp = xn
<a name="l00108"></a>00108         xr = np.zeros(b.shape)  <span class="comment"># USE INSIDE LOOP</span>
<a name="l00109"></a>00109         dim = diag0.size
<a name="l00110"></a>00110 
<a name="l00111"></a>00111         error = .10
<a name="l00112"></a>00112         <span class="keywordflow">while</span> error &gt; self.get_tolerance():
<a name="l00113"></a>00113             xr = xn.copy()
<a name="l00114"></a>00114             xn[0] = (1. - omega) * xp[0] + omega * (b[0] - diag1[0]
<a name="l00115"></a>00115                     * xp[1]) / diag0[0]
<a name="l00116"></a>00116             xn[0] = max(xn[0], const[0])
<a name="l00117"></a>00117             xn[1] = (1. - omega) * xp[1] + omega * (b[1] - diag_1[0]
<a name="l00118"></a>00118                     * xp[0] - diag1[1] * xp[2]) / diag0[1]
<a name="l00119"></a>00119             xn[1] = max(xn[1], const[1])
<a name="l00120"></a>00120 
<a name="l00121"></a>00121             <span class="keywordflow">for</span> i <span class="keywordflow">in</span> np.arange(2, dim - 2):
<a name="l00122"></a>00122                 tmp = diag_1[i - 1] * xp[i - 1] + diag1[i] * xp[i + 1]
<a name="l00123"></a>00123                 xn[i] = (1. - omega) * xp[i] + omega * (b[i] - tmp) \
<a name="l00124"></a>00124                     / diag0[i]
<a name="l00125"></a>00125                 xn[i] = max(xn[i], const[i])
<a name="l00126"></a>00126 
<a name="l00127"></a>00127             tmp = diag_1[dim - 3] * xp[-3] + diag1[dim - 2] * xp[-1]
<a name="l00128"></a>00128             xn[-2] = (1. - omega) * xp[-2] + omega * (b[-2] - tmp) \
<a name="l00129"></a>00129                 / diag0[-2]
<a name="l00130"></a>00130             xn[-2] = max(xn[-2], const[-2])
<a name="l00131"></a>00131 
<a name="l00132"></a>00132             tmp = diag_1[dim - 2] * xp[-2]
<a name="l00133"></a>00133             xn[-1] = (1. - omega) * xp[-1] + omega * (b[-1] - tmp) \
<a name="l00134"></a>00134                 / diag0[-1]
<a name="l00135"></a>00135             xn[-1] = max(xn[-1], const[-1])
<a name="l00136"></a>00136             error = np.linalg.norm((xr - xn).flatten())
<a name="l00137"></a>00137 
<a name="l00138"></a>00138     <span class="keyword">def </span>solve_lc_pendia(self):
<a name="l00139"></a>00139         <span class="keywordflow">if</span> <span class="keywordtype">None</span> == self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a>:
<a name="l00140"></a>00140             self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a0bd7ed48c6a7a39e945be4009251b731">omega</a> = self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a10bb69b8ef69ccea498ffbc372f95df6">_relaxationParameter</a>()
<a name="l00141"></a>00141 
<a name="l00142"></a>00142         M = self.get_matrix()
<a name="l00143"></a>00143         diag0 = diag(M)
<a name="l00144"></a>00144         (diag1, diag2) = (diag(M, 1), diag(M, 2))
<a name="l00145"></a>00145         (diag_1, diag_2) = (diag(M, -1), diag(M, -2))
<a name="l00146"></a>00146 
<a name="l00147"></a>00147         b = self.get_rhs()
<a name="l00148"></a>00148         xn = self.solution()
<a name="l00149"></a>00149 
<a name="l00150"></a>00150         const = self.get_constraint()
<a name="l00151"></a>00151         <span class="keywordflow">if</span> <span class="stringliteral">&#39;pSor&#39;</span> == self.get_lcp_solver_type():
<a name="l00152"></a>00152             <span class="keyword">assert</span> const != <span class="keywordtype">None</span>
<a name="l00153"></a>00153         <span class="keywordflow">else</span>:
<a name="l00154"></a>00154             const = xn
<a name="l00155"></a>00155 
<a name="l00156"></a>00156         omega = self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a>
<a name="l00157"></a>00157         xp = xn
<a name="l00158"></a>00158         xr = np.zeros(b.shape)  <span class="comment"># USE INSIDE LOOP</span>
<a name="l00159"></a>00159         dim = diag0.size
<a name="l00160"></a>00160 
<a name="l00161"></a>00161         error = .10
<a name="l00162"></a>00162         <span class="keywordflow">while</span> error &gt; self.get_tolerance():
<a name="l00163"></a>00163             xr = xn.copy()
<a name="l00164"></a>00164             xn[0] = (1. - omega) * xp[0] + omega * (b[0] - diag1[0]
<a name="l00165"></a>00165                     * xp[1] - diag2[0] * xp[2]) / diag0[0]
<a name="l00166"></a>00166             xn[0] = max(xn[0], const[0])
<a name="l00167"></a>00167             xn[1] = (1. - omega) * xp[1] + omega * (b[1] - diag_1[0]
<a name="l00168"></a>00168                     * xp[0] - diag1[1] * xp[2] - diag2[1] * xp[3]) \
<a name="l00169"></a>00169                 / diag0[1]
<a name="l00170"></a>00170             xn[1] = max(xn[1], const[1])
<a name="l00171"></a>00171 
<a name="l00172"></a>00172             <span class="keywordflow">for</span> i <span class="keywordflow">in</span> np.arange(2, dim - 2):
<a name="l00173"></a>00173                 tmp = diag_2[i - 2] * xp[i - 2] + diag_1[i - 1] * xp[i
<a name="l00174"></a>00174                         - 1] + diag1[i] * xp[i + 1] + diag2[i] * xp[i
<a name="l00175"></a>00175                         + 2]
<a name="l00176"></a>00176                 xn[i] = (1. - omega) * xp[i] + omega * (b[i] - tmp) \
<a name="l00177"></a>00177                     / diag0[i]
<a name="l00178"></a>00178                 xn[i] = max(xn[i], const[i])
<a name="l00179"></a>00179 
<a name="l00180"></a>00180             tmp = diag_2[dim - 4] * xp[-4] + diag_1[dim - 3] * xp[-3] \
<a name="l00181"></a>00181                 + diag1[dim - 2] * xp[-1]
<a name="l00182"></a>00182             xn[-2] = (1. - omega) * xp[-2] + omega * (b[-2] - tmp) \
<a name="l00183"></a>00183                 / diag0[-2]
<a name="l00184"></a>00184             xn[-2] = max(xn[-2], const[-2])
<a name="l00185"></a>00185 
<a name="l00186"></a>00186             tmp = diag_2[dim - 3] * xp[-3] + diag_1[dim - 2] * xp[-2]
<a name="l00187"></a>00187             xn[-1] = (1. - omega) * xp[-1] + omega * (b[-1] - tmp) \
<a name="l00188"></a>00188                 / diag0[-1]
<a name="l00189"></a>00189             xn[-1] = max(xn[-1], const[-1])
<a name="l00190"></a>00190             error = np.linalg.norm((xr - xn).flatten())
<a name="l00191"></a>00191 
<a name="l00192"></a>00192     <span class="keyword">def </span>solve(self):
<a name="l00193"></a>00193         <span class="keywordflow">if</span> <span class="keywordtype">None</span> == self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a>:
<a name="l00194"></a>00194             self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a0bd7ed48c6a7a39e945be4009251b731">omega</a> = self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a10bb69b8ef69ccea498ffbc372f95df6">_relaxationParameter</a>()
<a name="l00195"></a>00195 
<a name="l00196"></a>00196         <span class="keywordflow">if</span> 5 == self.get_matrix_attr():
<a name="l00197"></a>00197             <span class="keywordflow">return</span> self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a41cdc717a4b4be2c66fea81074a32016">solve_lc_pendia</a>()
<a name="l00198"></a>00198 
<a name="l00199"></a>00199         <span class="keywordflow">if</span> 3 == self.get_matrix_attr():
<a name="l00200"></a>00200             <span class="keywordflow">return</span> self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#a10b8a0c8ebd082217d76c15f2e7b9161">solve_lc_tridia</a>()
<a name="l00201"></a>00201 
<a name="l00202"></a>00202         b = self.get_rhs()
<a name="l00203"></a>00203         M = self.get_matrix()
<a name="l00204"></a>00204         xn = self.solution()
<a name="l00205"></a>00205 
<a name="l00206"></a>00206         const = self.get_constraint()
<a name="l00207"></a>00207         <span class="keywordflow">if</span> <span class="stringliteral">&#39;pSor&#39;</span> == self.get_lcp_solver_type():
<a name="l00208"></a>00208             <span class="keyword">assert</span> const != <span class="keywordtype">None</span>
<a name="l00209"></a>00209         <span class="keywordflow">else</span>:
<a name="l00210"></a>00210             const = xn
<a name="l00211"></a>00211 
<a name="l00212"></a>00212         omega = self.<a class="code" href="classsolvers_1_1SorSolvers_1_1Sor.html#af838f1e4fce19ba6bb2826b8a07bbf76">m_omega</a>
<a name="l00213"></a>00213         xp = xn
<a name="l00214"></a>00214         xr = np.zeros(b.shape)  <span class="comment"># USE INSIDE LOOP</span>
<a name="l00215"></a>00215         error = .10
<a name="l00216"></a>00216         <span class="keywordflow">while</span> error &gt; self.get_tolerance():
<a name="l00217"></a>00217             xr = xn.copy()
<a name="l00218"></a>00218             <span class="keywordflow">for</span> i <span class="keywordflow">in</span> np.arange(xp.size):
<a name="l00219"></a>00219                 xn[i] = (1. - omega) * xp[i] + omega * (xp[i] + (b[i]
<a name="l00220"></a>00220                         - np.dot(M[i, :], xp)) / M[i, i])
<a name="l00221"></a>00221                 xn[i] = max(xn[i], const[i])
<a name="l00222"></a>00222             error = np.linalg.norm((xr - xn).flatten())
<a name="l00223"></a>00223 
<a name="l00224"></a>00224 
</pre></div></div>
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